We consider the neural sparse representation to solve Boltzmann equation with BGK and quadratic collision model, where a network-based ansatz that can approximate the distribution function with extremely high efficiency is proposed. Precisely, fully connected neural networks are employed in the time and spatial space so as to avoid the discretization in space and time. The different low-rank representations are utilized in the microscopic velocity for the BGK and quadratic collision model, resulting in a significant reduction in the degree of freedom. We approximate the discrete velocity distribution in the BGK model using the canonical polyadic decomposition. For the quadratic collision model, a data-driven, SVD-based linear basis is built based on the BGK solution. All these will significantly improve the efficiency of the network when solving Boltzmann equation. Moreover, the specially designed adaptive-weight loss function is proposed with the strategies as multi-scale input and Maxwellian splitting applied to further enhance the approximation efficiency and speed up the learning process. Several numerical experiments, including 1D wave and Sod problems and 2D wave problem, demonstrate the effectiveness of these neural sparse representation methods.

翻译：本文考虑利用神经稀疏表示方法求解包含BGK和二次碰撞模型的玻尔兹曼方程，提出了一种基于网络的高效近似分布函数的解空间表达。具体而言，在时间和空间维度采用全连接神经网络以避免离散化；针对BGK和二次碰撞模型，在微观速度维度分别采用不同的低秩表示以显著降低自由度。对于BGK模型，我们利用规范多项式分解近似离散速度分布；对于二次碰撞模型，则基于BGK解构建数据驱动的SVD线性基底。这些措施极大提升了网络求解玻尔兹曼方程的效率。此外，本文设计了自适应权重损失函数，并结合多尺度输入与麦克斯韦分裂策略，进一步增强了近似效率并加速了学习过程。一维波动方程与Sod问题、二维波动方程的数值实验验证了所提神经稀疏表示方法的有效性。